Interdisciplinary Applied Mathematics

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4.2.3 Summary


In Section 4.2.2, we developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for pipes and rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations in section 4.2.1. The general velocity slip boundary condition (equation (2.43)) and the rarefaction correction factor (equation (4.23)) are the two primary components of this unified model.


The general slip boundary condition (equation (2.43)) gives the correct nondimensional velocity profile, where the normalization is obtained using the local channel averaged velocity. This eliminates the flowrate dependence in modeling the velocity profile. For channel flows, using equation (2.39), we obtain b = —1 in the slip flow regime. Evidence based on comparisons of the model with the DSMC and Boltzmann solutions shows that b = —1 in the entire Knudsen regime.


In order to model the flowrate variations with respect to the Knudsen number Kn, we introduced the rarefaction correction factor as Cr = 1 + a Kn. This form of the correction factor was justified using two independent arguments: first, the apparent diffusion coefficient; and second, the ratio of intermolecular collisions to the total molecular collisions. We must note that a cannot be a constant. Physical considerations to match the slip flowrate require a ^ 0 for Kn < 0.1, while a ^ a0 in the free molecular flow regime. The variation of between zero and a known a0 value is approximated using equation (4.34), which introduced two empirical parameters ai and в to the new model.

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